Problem: Donna has $n$ boxes of doughnuts. Each box contains $13$ doughnuts.

After eating one doughnut, Donna is able to rearrange the remaining doughnuts into bags so that each bag contains $9$ doughnuts, and none are left over. What is the smallest possible value of $n$?
At the beginning, there are $13n$ doughnuts. After $1$ doughnut is eaten, the number of remaining doughnuts is a multiple of $9$. Therefore, the original number of doughnuts was $1$ more than a multiple of $9$. Expressing this as a congruence, we have $$13n\equiv 1\pmod 9,$$or in other words, $n\equiv 13^{-1}\pmod 9$. Since $13\equiv 4\pmod 9$, we can also write $n\equiv 4^{-1}\pmod 9$.

Because $4\cdot 7=28\equiv 1$, we have $4^{-1}\equiv 7\pmod 9$. Therefore, $n\equiv 7\pmod 9$. We know $n$ must be a nonnegative integer, so the smallest possible value of $n$ is $\boxed{7}$.

We can check our answer: If $n=7$, then Donna started with $7\cdot 13=91$ doughnuts; after eating one, she had $90$, which is a multiple of $9$.